Small-world MCMC and convergence to multi-modal distributions: From slow mixing to fast mixing

We compare convergence rates of Metropolis--Hastings chains to multi-modal target distributions when the proposal distributions can be of ``local'' and ``small world'' type. In particular, we show that by adding occasional long-range jumps to a given local proposal distribution, one can turn a chain that is ``slowly mixing'' (in the complexity of the problem) into a chain that is ``rapidly mixing.'' To do this, we obtain spectral gap estimates via a new state decomposition theorem and apply an isoperimetric inequality for log-concave probability measures. We discuss potential applicability of our result to Metropolis-coupled Markov chain Monte Carlo schemes.Comment: Published at http://dx.doi.org/10.1214/105051606000000772 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Spatial opinion dynamics and the effects of two types of mixing

Spatially situated opinions that can be held with different degrees of conviction lead to spatiotemporal patterns such as clustering (homophily), polarization, and deadlock. Our goal is to understand how sensitive these patterns are to changes in the local nature of interactions. We introduce two different mixing mechanisms, spatial relocation and nonlocal interaction (“telephoning”), to an earlier fully spatial model (no mixing). Interestingly, the mechanisms that create deadlock in the fully spatial model have the opposite effect when there is a sufficient amount of mixing. With telephoning, not only is polarization and deadlock broken up, but consensus is hastened. The effects of mixing by relocation are even more pronounced. Further insight into these dynamics is obtained for selected parameter regimes via comparison to the mean-field differential equations

Application of Ecological Network Theory to the Human Microbiome

In healthy humans, many microbial consortia constitute rich ecosystems with dozens to hundreds of species, finely tuned to functions relevant to human health. Medical interventions, lifestyle changes, and the normal rhythms of life sometimes upset the balance in microbial ecosystems, facilitating pathogen invasions or causing other clinically relevant problems. Some diseases, such as bacterial vaginosis, have exactly this sort of community etiology. Mathematical network theory is ideal for studying the ecological networks of interacting species that comprise the human microbiome. Theoretical networks require little consortia specific data to provide insight into both normal and disturbed microbial community functions, but it is easy to incorporate additional empirical data as it becomes available. We argue that understanding some diseases, such as bacterial vaginosis, requires a shift of focus from individual bacteria to (mathematical) networks of interacting populations, and that known emergent properties of these networks will provide insights that would be otherwise elusive

Pubertal presentation in seven patients with congenital adrenal hyperplasia due to P450 Oxidoreductase deficiency

Context: P450 oxidoreductase (POR) is a crucial electron donor to all microsomal P450 cytochrome (CYP) enzymes including 17α-hydroxylase (CYP17A1), 21-hydroxylase (CYP21A2) and P450 aromatase. Mutant POR causes congenital adrenal hyperplasia with combined glucocorticoid and sex steroid deficiency. P450 oxidoreductase deficiency (ORD) commonly presents neonatally, with disordered sex development in both sexes, skeletal malformations, and glucocorticoid deficiency. \ud \ud Objective: The aim of the study was to describe the clinical and biochemical characteristics of ORD during puberty. \ud \ud Design: Clinical, biochemical, and genetic assessment of seven ORD patients (five females, two males) presenting during puberty was conducted. \ud \ud Results: Predominant findings in females were incomplete pubertal development (four of five) and large ovarian cysts (five of five) prone to spontaneous rupture, in some only resolving after combined treatment with estrogen/progestin, GnRH superagonists, and glucocorticoids. Pubertal development in the two boys was more mildly affected, with some spontaneous progression. Urinary steroid profiling revealed combined CYP17A1 and CYP21A2 deficiencies indicative of ORD in all patients; all but one failed to mount an appropriate cortisol response to ACTH stimulation indicative of adrenal insufficiency. Diagnosis of ORD was confirmed by direct sequencing, demonstrating disease-causing POR mutations. \ud \ud Conclusion: Delayed and disordered puberty can be the first sign leading to a diagnosis of ORD. Appropriate testosterone production during puberty in affected boys but manifest primary hypogonadism in girls with ORD may indicate that testicular steroidogenesis is less dependent on POR than adrenal and ovarian steroidogenesis. Ovarian cysts in pubertal girls may be driven not only by high gonadotropins but possibly also by impaired CYP51A1-mediated production of meiosis-activating sterols due to mutant POR

Representations for continuous additive functionals of super-Brownian and super-stable processes

We study a class of continuous additive functionals for super-Brownian and super-stable processes. These are given in terms of a Tanaka-like formula that generalizes the one for local times. We give representations for these additive functionals in terms of the corresponding local times. As an example, we discuss fractional Laplacians of super-Brownian local times.Superprocess Measure-valued process Additive functional Local time Fractional derivative Tanaka formula

Conditioned superprocesses and their weighted occupation times

A superprocess which starts at a finite measure will die in finite time. We consider a class of measure-valued Markov processes obtained by conditioning such a process to stay alive forever. More specifically, we study the asymptotic behavior of the weighted occupation times for these "conditioned superprocesses". We give necessary and sufficient conditions, based on the asymptotics of the underlying motion, for the total occupation time to be infinite. Some special cases are investigated. We also prove that, when properly scaled, the occupation times for conditioned super-Brownian motion converge to a function of the local time when d [less-than-or-equals, slant] 3.Superprocesses Measure-valued processes Branching processes Conditioned superprocesses Weighted occupation times Harris processes Local times

Spatial self-organization in a cyclic resource-species model

Biological communities are remarkable in their ability to form cooperative ensembles that lead to coexistence through various types of niche partitioning, usually intimately tied to spatial structure. This is especially true in microbial settings where differential expression and regulation of genes allows members of a given species to alter their lifestyle so as to fill a functional role within the community. The resulting species interactions can involve feedback, as in the case of some bacterial consortia that participate in the cooperative degradation of a given resource in a succession of steps and in such a way that certain "later" species provide catalytic support for the primary degrader. We seek to capture the essential features of such spatially extended biological systems by introducing a lattice-based stochastic spatial model (interacting particle system) with cyclic local dynamics. Here, a given site progresses through a sequence of resource and species states in a prescribed order. Furthermore, this succession of states (at a site) is assumed to form a cyclic pattern due to a natural feedback mechanism. We explore conditions under which all the species are able to coexist and consider the extent to which this coexistence requires the development of spatio-temporal patterns, including spiral waves. This self-organization, if it occurs, results when synchronization of the dynamics at the microscopic level leads to macroscopic patterns. These patterns result in consumer-driven resource fluctuations that generate a form of spatio-temporal niche partitioning. As with most models of this complexity, we employ a mixture of mathematical analysis and simulations to develop an understanding of the resulting dynamics